As a native English speaker, which two further languages would give me the most travel utility worldwide?

To provide an extension to jpatokal's answer and MeNoTalk's comment, I would argue for learning several romance languages. In raw number of speakers, I think Spanish is clearly ahead but French is also useful in large parts of the world (even if only spoken as a second language by a limited number of people, it is an official language and the language of education in many countries).

Most importantly, for you as an English speaker, it should not be too difficult to learn French, Spanish and maybe Portuguese or even Italian as well. While English is most closely related to Germanic languages, it also has a large latin-based lexicon that will help you learn romance languages. Once you know one of these languages, learning two or three should be even quicker.

As a European French speaker I could be biased of course but I have known (native) speakers of Spanish or French becoming fluent in the other language in a matter of months. I also know many people who are fluent in four or five romance languages. So in terms of reward, you could communicate with many people in South America, Africa, large parts of Europe and a few other countries elsewhere for a relatively small amount of effort.

By contrast, learning Arabic (some overlap with French here, incidentally) and Chinese is arguably more interesting as both of these languages have many speakers and would open a completely different set of cultures to you but you would be looking at many years of effort to become even moderately competent in one of them and then have to start almost from scratch for the second one.

You have to consider Malay (or Indonesian. It is mostly the same).

Length of time required to be "functional" in the language
From this point of view, it is by far the easiest language I've ever tried (I've never tried Esperanto).
Pronunciation and grammar are easy. Writing even more (Mostly similar to Italian)

Chance of reaching proficiency within any meaningful timeframe
I had not reached this point yet(3 years), but it seems reasonable. I would say 3 years if you learn seriously (I mean native proficiency).

Utility of the Language
Main language of Indonesia, Malaysia and Brunei.
important in Singapore (most Chinese understand Malay), south Thailand and a few other
Can be useful in Taiwan (It is easier to find an Indonesian migrant than an English speaking Taiwanese)
All in all, between 220 and 280 million speakers.

Regional dialects destroying proficiency
Indonesian Malay is 99% understood in Malaysia, Brunai and Singapore (among Malay speaker). The Malay of South Thailand is probably significantly different.

Easier Secondary languages underpinning the society (Arabic versus French argument)
Not sure I understand what you mean.
Chinese (Hokian and Mandarin) are secondary languages in most of the area, but you'll never learn those in your lifetime (assuming you are native English speaker and beyond 5 year old)

Similarity of Language families
Similar to Tagalog (aka Filipino)
Possibly similar to other Austronesian languages. I recognized some Amis words while in Taiwan.

Enjoyment is a minor consideration Good. Because Indonesian pop music is a kind of torture (I do like Batak music, however)

You guys are forgetting one of the most important languages in the world now, and for the future - Portuguese !

Once you become fluent in Portuguese you get Spanish practically for free, at in terms of understanding it. The grammar, vocabulary and overall structure between Portuguese and Spanish is about 90%. Not kidding here.

In terms of importance as a major language of the world, there are many, many reasons. Portuguese is spoken officially by 280 million people, on 5 continents, and it is 5th most spoken language in the world! Portuguese is spoken officially in: Portugal (inc. Azores and Madeira), Brazil, Angola, Mozambique, Guinea-Bissau, Cape Verde, San Tome and Principe Islands, East Timor, Macau, Equatorial Guinea.

Brazil has at moment the 5th strongest economy in the world! Brazil is a HUGE country which occupies roughly 50% of the land area in South America. It is an extremely beautiful, and natural resource rich country which has an abundance of: oil, natural gas, gold, silver, coffee, sugar, and is a world leader in aerospace, telecommunications, computer technology, and a major producer/exporter of important consumer goods and foods. 51% of all the speakers in South America speaks Portuguese.

Portuguese is he 3rd most spoken European language in the world, the 3rd most spoken in the Americas, and the 1st most spoken language in the Southern Hemisphere.

Portuguese is the language of many important global organizations including: Mercosur, Latin American Union, European Union, African Union, and CPLP (Community of Portuguese Speaking Countries - which even have their own Olympic Games).

Portuguese is spoken in 6 African countries, whereby Angola an Mozambique have the top 5 strongest African economies in Africa. Both of these countries have abundant natural resources including: oil, natural gas, gold, diamonds, silver, sugar, coffee, etc.

The main trading partners of all of the Portuguese speaking nations, especially Brazil, Angola, Portugal, Mozambique, are: China, United States, India, Canada, and Australia among many others. As such, the Portuguese language is well positioned as an important language of the 21st century and beyond world economy.

Portuguese is currently being taught as a preferred 2d language in many countries in the world including: Japan, China, Argentina, Uruguay, Paraguay, Venezuela, South Africa, Namibia, Senegal, France, Spain, Luxembourg, Andorra, etc.

There are huge Portuguese immigrant communities all over the world in the following countries: Canada, United States, Venezuela, England, France, Germany, Luxembourg, Andorra, Namibia, South Africa, Israel, Australia, India, China, Japan, etc.

And Portuguese is soon to become the 7th official language of the United Nations.

All of the above reasons given are proof positive of why fluency, or even basic conversational Portuguese is a huge asset in today's globalized world.

And most importantly, Portuguese is a very beautiful and expressive romance language, which is becoming more and more important everyday! And knowing this wonderful language means that you will be able to communicate with millions of Portuguese speakers all over the world. Furthermore, knowledge of the Portuguese language will give you a huge advantage over everyone else if you ever visit a Spanish speaking country!

As a English speaker if you want to maximize the places that you can visit your obvious first elections should be Spanish and Russian, at least in my experience. Of course I have a Bias since I am native Spanish and I speak some Russian, but I will try to be objective.

Whereas some could argue that Arabic is more widely spoke than Russian I think that are several arguments against it. First, the pronunciation is practically impossible for a non native. Second, there is a lot of dialects, than for a nonnative will be hard to understand.

The arguments in favor of Spanish are the very big amount of native speakers. Lot of times, traveling across Europe I have found native Spanish speakers working in different places or just living there. In places so different as Kiev, Berlin Oslo, Prague, Zurich or Moscow. Also I have found a surprisingly amount of people who have learnt Spanish at some point in Germany. The second argument is of course South American Countries. And the third is that you can understand people from Italy and Portuguese countries if they speak slow. You will not have a philosophical debate but you will understand if they want to overtrick you and also any kind of explanation that they could give.

Arguments in favor of Russian is that is spoken in all post soviet republics and also in some more countries like Poland, Baltic countries (only old people). Very important fact about Russian is that is hardly substitute for any other language. Yes you can travel across Russia without knowing Russian but is hard as hell, I did in the past an I do not recommend the experience. In small cities you can easily find that nobody speaks English.

The same argument it can be used in favor of Spanish. Go to Spain knowing only English is much harder than go to Portugal knowing only English.

Probably my answer is very focus in the western world, since I do not know a lot about Asian languages, but I hope it can be helpful since it is based in m experience and not in my opinion.